We like to ask for the comparison of two topologies of three 2-tori inside the same 4-manifolds glued from two different diffeomorphisms (see the end).

Given an embedded torus $T$ with trivial normal bundle in a 4–manifold $X$, a torus surgery on $T$ (also called a logarithmic transform) is the process of removing a neighborhood $\nu T$, which will be denoted as $(X - \nu T)$ and re-gluing $D^2 \times T^2$ by some diffeomorphism $\phi$ of the boundary to attain

$$X_T = (X - \nu T) \cup_{\phi} D^2 \times T^2,$$

The cup product is the gluing, and the gluing is under the diffeomorphism $\phi$.

We can choose the following manifolds,

we call $X \equiv S^4$ and call $A \equiv D^2 \times T^2$

$$(X-A) \cup_{\phi} A \equiv (S^4 - D^2 \times T^2) \cup_{\phi} D^2 \times T^2,$$

We denote $T^3=S^1_x \times S^1_y \times S^1_z$, denote each $S^1$ as unit circles, where the $S^1_x$ bounds $D^2_{xr}$, and $T^2_{yz}$ are parametrized by $S^1_y \times S^1_z$. I am interested in choosing the two diffeomorphism ${\phi}$ maps, say ${\phi}_{xy}$ and ${\phi}_{xyz}$ as permuting the three circles of $T^3=\partial(D^2_{xr} \times T^2_{yz})$ on the glued boundary in two different ways:

(1) Say, $\phi_{xy}$ permutes only the two $S^1_x$ and $S^1_y$ circles: $$ \phi_{xy}=\begin{pmatrix} 0 & -1 & 0\\ 1 & 0 & 0\\ 0 & 0 & 1 \end{pmatrix}, $$ (2) Say, $\phi_{xyz}$ cyclic permutes all $S^1_x$, $S^1_y$, $S^1_z$ circles: $$ \phi_{xyz}=\begin{pmatrix} 0 & 0 & 1\\ 1 & 0 & 0\\ 0 & 1 & 0 \end{pmatrix} $$ here both the matrix representations $\phi_{xy}$ and $\phi_{xyz}$ transform the coordinates $x,y,z$ between two $T^3$ boundaries of $S^1_x \times S^1_y \times S^1_z$.

One can check that both gluing $$(X - A) \cup_{\phi_{xy}} A = (X - A) \cup_{\phi_{xyz}} A =S^3 \times S^1 \# S^2 \times S^2$$ give the same outcome manifold.

Question: Now we insert a $T^2$-torus into A along the generator of homology group $H^2(A, \mathbb{Z})=\mathbb{Z}$, and insert two additional $T^2$-tori into $X-A$ along the generators of homology group $H^2(X-A \times T^2, \mathbb{Z})=\mathbb{Z}^2$ respectively. If we consider glue the two diffeomorphism $\phi_{xy}$ and $\phi_{xyz}$ of gluings to glue $X-A$ and $A$, then we imagine that the three $T^2$ tori are inside the final manifold $S^3 \times S^1 \# S^2 \times S^2$ for both two types of gluings: $\phi_{xy}$-gluing and the $\phi_{xyz}$-gluing. My question is that, comparing the two scenarios of gluings, namely $\phi_{xy}$-gluing and the $\phi_{xyz}$-gluing, are the three $T^2$-tori locate inside the final manifold $S^3 \times S^1 \# S^2 \times S^2$ in the same way in terms of their topology? Why?

  • $\begingroup$ My tentative understanding is that the $\phi_{xy}$-gluing and the $\phi_{xyz}$-gluing give the same topology of three $T^2$-tori inside $S^3 \times S^1 \# S^2 \times S^2$. But I don't know the formal argument. $\endgroup$ – wonderich Nov 6 '16 at 1:02

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